Theorems · Theorem · category theory

CategoryTheory.Limits.preserves_cokernel_iso_comp_cokernel_map

∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C]
  {D : Type u₂} [inst_2 : CategoryTheory.Category.{v₂, u₂} D] [inst_3 : CategoryTheory.Limits.HasZeroMorphisms D]
  (G : CategoryTheory.Functor C D) [inst_4 : G.PreservesZeroMorphisms] {X Y : C} (f : X ⟶ Y)
  [inst_5 : CategoryTheory.Limits.HasCokernel f] [inst_6 : CategoryTheory.Limits.HasCokernel (G.map f)]
  [inst_7 : CategoryTheory.Limits.PreservesColimit (CategoryTheory.Limits.parallelPair f 0) G] {X' Y' : C} (g : X' ⟶ Y')
  [inst_8 : CategoryTheory.Limits.HasCokernel g] [inst_9 : CategoryTheory.Limits.HasCokernel (G.map g)]
  [inst_10 : CategoryTheory.Limits.PreservesColimit (CategoryTheory.Limits.parallelPair g 0) G] (p : X ⟶ X')
  (q : Y ⟶ Y') (hpq : CategoryTheory.CategoryStruct.comp f q = CategoryTheory.CategoryStruct.comp p g),
  CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.PreservesCokernel.iso G f).hom
      (CategoryTheory.Limits.cokernel.map (G.map f) (G.map g) (G.map p) (G.map q) ⋯) =
    CategoryTheory.CategoryStruct.comp (G.map (CategoryTheory.Limits.cokernel.map f g p q hpq))
      (CategoryTheory.Limits.PreservesCokernel.iso G g).hom

Defined in

Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels

Cited by

1 results in Mathlib

Foundations

Depth 42 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.Limits.HasZeroMorphismsCategoryTheory.CategoryCategoryTheory.Limits.HasZeroMorphismsCategoryTheory.Functor.PreservesZeroMorphismsCategoryTheory.Limits.HasCokernelCategoryTheory.Limits.HasCokernelCategoryTheory.Limits.PreservesColimitCategoryTheory.Limits.HasCokernelCategoryTheory.Limits.HasCokernelCategoryTheory.Limits.PreservesColimit